vertices with similar weights not moving at same speed
Upon further playing around with the mesh, I found out that I should do the following to get what I want:


In edit mode, select the island of the cloth mesh, in this case, the t-shirt.
Press P to separate the cloth, click by "Selection".
Go to object mode, parent the rest of the mesh to the armature, with automatic weights. 
Then parent the cloth just separated to the armature too.
Go to pose mode and move the body, the cloth will follow it.
Go back to object mode, select the cloth then the body, and then join the two meshes, to get back the original mesh.
Going back to pose mode and moving the armature around still shows that the cloth follows the body's motion.


That's it, this worked for me and I hope someone will also find it helpful. If anyone has a different way of approaching this, please share.